具有相等Grundy支配和独立数的图

IF 0.9 4区 数学 Q3 MATHEMATICS, APPLIED Discrete Optimization Pub Date : 2023-05-01 DOI:10.1016/j.disopt.2023.100777
Gábor Bacsó , Boštjan Brešar , Kirsti Kuenzel , Douglas F. Rall
{"title":"具有相等Grundy支配和独立数的图","authors":"Gábor Bacsó ,&nbsp;Boštjan Brešar ,&nbsp;Kirsti Kuenzel ,&nbsp;Douglas F. Rall","doi":"10.1016/j.disopt.2023.100777","DOIUrl":null,"url":null,"abstract":"<div><p>The Grundy domination number, <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>gr</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, of a graph <span><math><mi>G</mi></math></span> is the maximum length of a sequence <span><math><mrow><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></math></span> of vertices in <span><math><mi>G</mi></math></span> such that for every <span><math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math></span>, the closed neighborhood <span><math><mrow><mi>N</mi><mrow><mo>[</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span> contains a vertex that does not belong to any closed neighborhood <span><math><mrow><mi>N</mi><mrow><mo>[</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>j</mi><mo>&lt;</mo><mi>i</mi></mrow></math></span>. It is well known that the Grundy domination number of any graph <span><math><mi>G</mi></math></span> is greater than or equal to the upper domination number <span><math><mrow><mi>Γ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, which is in turn greater than or equal to the independence number <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. In this paper, we initiate the study of the class of graphs <span><math><mi>G</mi></math></span> with <span><math><mrow><mi>Γ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>gr</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and its subclass consisting of graphs <span><math><mi>G</mi></math></span> with <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>gr</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. We characterize the latter class of graphs among all twin-free connected graphs, provide a number of properties of these graphs, and prove that the hypercubes are members of this class. In addition, we give several necessary conditions for graphs <span><math><mi>G</mi></math></span> with <span><math><mrow><mi>Γ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>gr</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and present large families of such graphs.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100777"},"PeriodicalIF":0.9000,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Graphs with equal Grundy domination and independence number\",\"authors\":\"Gábor Bacsó ,&nbsp;Boštjan Brešar ,&nbsp;Kirsti Kuenzel ,&nbsp;Douglas F. Rall\",\"doi\":\"10.1016/j.disopt.2023.100777\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The Grundy domination number, <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>gr</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, of a graph <span><math><mi>G</mi></math></span> is the maximum length of a sequence <span><math><mrow><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></math></span> of vertices in <span><math><mi>G</mi></math></span> such that for every <span><math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math></span>, the closed neighborhood <span><math><mrow><mi>N</mi><mrow><mo>[</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span> contains a vertex that does not belong to any closed neighborhood <span><math><mrow><mi>N</mi><mrow><mo>[</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>j</mi><mo>&lt;</mo><mi>i</mi></mrow></math></span>. It is well known that the Grundy domination number of any graph <span><math><mi>G</mi></math></span> is greater than or equal to the upper domination number <span><math><mrow><mi>Γ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, which is in turn greater than or equal to the independence number <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. In this paper, we initiate the study of the class of graphs <span><math><mi>G</mi></math></span> with <span><math><mrow><mi>Γ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>gr</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and its subclass consisting of graphs <span><math><mi>G</mi></math></span> with <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>gr</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. We characterize the latter class of graphs among all twin-free connected graphs, provide a number of properties of these graphs, and prove that the hypercubes are members of this class. In addition, we give several necessary conditions for graphs <span><math><mi>G</mi></math></span> with <span><math><mrow><mi>Γ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>gr</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and present large families of such graphs.</p></div>\",\"PeriodicalId\":50571,\"journal\":{\"name\":\"Discrete Optimization\",\"volume\":\"48 \",\"pages\":\"Article 100777\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-05-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Optimization\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1572528623000191\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Optimization","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1572528623000191","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 1

摘要

图G的Grundy支配数γgr(G)是G中顶点序列(v1,v2,…,vk)的最大长度,使得对于每i∈{2,…,k},闭邻域N[vi]包含不属于任何闭邻域N[fj]的顶点,其中j<;i.众所周知,任何图G的Grundy控制数都大于或等于上控制数Γ(G),而上控制数又大于或等于独立数α(G)。本文研究了Γ(G)=γgr(G)的图G的一类及其由α(G)=γgr(G)的图组成的子类。我们刻画了所有双自由连通图中的后一类图,给出了这些图的一些性质,并证明了超立方体是这类图的成员。此外,我们给出了Γ(G)=γgr(G)的图G的几个必要条件,并给出了这类图的大族。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
Graphs with equal Grundy domination and independence number

The Grundy domination number, γgr(G), of a graph G is the maximum length of a sequence (v1,v2,,vk) of vertices in G such that for every i{2,,k}, the closed neighborhood N[vi] contains a vertex that does not belong to any closed neighborhood N[vj], where j<i. It is well known that the Grundy domination number of any graph G is greater than or equal to the upper domination number Γ(G), which is in turn greater than or equal to the independence number α(G). In this paper, we initiate the study of the class of graphs G with Γ(G)=γgr(G) and its subclass consisting of graphs G with α(G)=γgr(G). We characterize the latter class of graphs among all twin-free connected graphs, provide a number of properties of these graphs, and prove that the hypercubes are members of this class. In addition, we give several necessary conditions for graphs G with Γ(G)=γgr(G) and present large families of such graphs.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Discrete Optimization
Discrete Optimization 管理科学-应用数学
CiteScore
2.10
自引率
9.10%
发文量
30
审稿时长
>12 weeks
期刊介绍: Discrete Optimization publishes research papers on the mathematical, computational and applied aspects of all areas of integer programming and combinatorial optimization. In addition to reports on mathematical results pertinent to discrete optimization, the journal welcomes submissions on algorithmic developments, computational experiments, and novel applications (in particular, large-scale and real-time applications). The journal also publishes clearly labelled surveys, reviews, short notes, and open problems. Manuscripts submitted for possible publication to Discrete Optimization should report on original research, should not have been previously published, and should not be under consideration for publication by any other journal.
期刊最新文献
Anchor-robust project scheduling with non-availability periods Corrigendum to “Bilevel time minimizing transportation problem” [Discrete Optim.] 5 (4) (2008) 714–723 Circuit and Graver walks and linear and integer programming Approximation schemes for Min-Sum k-Clustering Easy and hard separation of sparse and dense odd-set constraints in matching
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1